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Current Electricity question

2022 · 27 Jul · Shift 1 · Q57
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  5. /2022 · 27 Jul · Shift 1 · Q57

Current Electricity question

2022 · 27 Jul · Shift 1 · Q57

JEE MainPhysicsCurrent ElectricityMCQ+4 / −1
Two sources of equal emfs are connected in series. This combination is connected to an external resistance R. The internal resistances of the two sources are r1r_{1}r1​ and r2r_{2}r2​ (r1>r2)\left(r_{1}>r_{2}\right)(r1​>r2​). If the potential difference across the source of internal resistance r1r_{1}r1​ is zero, then the value of R will be :
  1. A
    r1−r2r_{1}-r_{2}r1​−r2​
  2. B
    r1r2r1+r2\frac{r_{1} r_{2}}{r_{1}+r_{2}}r1​+r2​r1​r2​​
  3. C
    r1+r22\frac{r_{1}+r_{2}}{2}2r1​+r2​​
  4. D
    r2−r1r_{2}-r_{1}r2​−r1​
View written solutionFree

Correct answer: A

  1. Set up the circuit

Let each source have emf EEE. Since the two equal emfs are connected in series, the total emf is 2E2E2E and the total internal resistance is (r1+r2).(r_1+r_2).(r1​+r2​).

This combination is connected to external resistance RRR, so total resistance in the circuit is R+r1+r2.R+r_1+r_2.R+r1​+r2​.

Hence the circuit current is I=2ER+r1+r2.I=\frac{2E}{R+r_1+r_2}.I=R+r1​+r2​2E​.

  1. Condition given in the question

The potential difference across the source with internal resistance r1r_1r1​ is zero.

For a cell delivering current, terminal potential difference is V=E−Ir.V = E - Ir.V=E−Ir.

So for the source with internal resistance r1r_1r1​, E−Ir1=0.E-Ir_1=0.E−Ir1​=0. Thus, I=Er1.I=\frac{E}{r_1}.I=r1​E​.

  1. Equate the two expressions for current

From the full circuit, I=2ER+r1+r2I=\frac{2E}{R+r_1+r_2}I=R+r1​+r2​2E​

and from the zero terminal voltage condition, I=Er1.I=\frac{E}{r_1}.I=r1​E​.

Therefore, 2ER+r1+r2=Er1.\frac{2E}{R+r_1+r_2}=\frac{E}{r_1}.R+r1​+r2​2E​=r1​E​.

Cancel EEE: 2R+r1+r2=1r1.\frac{2}{R+r_1+r_2}=\frac{1}{r_1}.R+r1​+r2​2​=r1​1​.

Cross-multiplying, 2r1=R+r1+r2.2r_1=R+r_1+r_2.2r1​=R+r1​+r2​.

So, R=r1−r2.R=r_1-r_2.R=r1​−r2​.

  1. Check with options

The value of RRR is r1−r2\boxed{r_1-r_2}r1​−r2​​ which corresponds to Option A.

Since r1>r2r_1>r_2r1​>r2​, this is positive and physically valid.

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